Summary: A method is introduced to represent many-body systems of arbitrary dimensionality by planar curves. The positions and momenta of the particles are the parameters of a time-dependent nonlinear transformation, which maps the many-body dynamics of the real system to the motion of the curve. The description of the system as a point in a multidimensional phase space is thus replaced by a two-dimensional continuous line. Expressions for the curvature along the curve and the dynamic structure factor are obtained. The formulation holds for Hamiltonian and non-Hamiltonian systems, and two explicit examples are analyzed: harmonic oscillators and a quadratic system.